Medically, ask your pediatrician. I'd presume that for such a "short" period of time (not talking 3 months at base-camp, correct?), a child would have the same oxygen requirements as an adult (except for pre-existing conditions). If in doubt... as a peditrician.

Personally, it sounds to be a nightmare. Start approaching FL120, and the ear-event will not be pretty.

Please describe the anticipated airframe, number of engines, and anticipated maximimum flight ceiling (airline and flight number would help).